Sometime ago I overheard a conversation in which someone said "studying functions fields is the same thing as studying algebraic curves".
After looking it up, I've found these two results (I'll always assume $k$ algebrically closed):
i) Any two projective curves $X,Y$ are birrationaly equivalent $\Leftrightarrow k(X)\simeq k(Y)$
ii) Two non-singular projective curves $X,Y$ are isomorphic $\Leftrightarrow X,Y$ are birrationally equivalent.
From these two, the phrase above doesn't seem accurate, because given a function field, it's not obvious that it correspond to $k(X)$ for some $X$. Besides, there is the issue of non-singularity of $X$, which I don't know how to deal with.
They were probably referring to a certain equivalence of categories. Let me (try to) formulate it for $k = \mathbb{C}$ if I manage to recall correctly:
The following categories are equivalent:
i) The category of algebraic curves with dominant rational maps
ii) The category of smooth projective curves with dominant morphisms
iii) The category of riemann surfaces with non-constant holomorphic maps
vi) The category of finitely generated (as $\mathbb{C}$-algebra) field extensions $\mathbb{C} \subset L$ of $\text{trdeg}_{\mathbb{C}}(L) = 1$ with homomorphisms of $\mathbb{C}$-algebras
We can also state vi) as
vi') The category of function fields in one variable with homomorphisms of $\mathbb{C}$-algebras
I think you can find that statement (or a similar one) in Introduction to Compact Riemann Surfaces and Dessins d'Enfants by Girondo and González-Diez.